A note on the coefficient array of a generalized Fibonacci polynomial (original) (raw)

On generalized Fibonacci and Lucas polynomials

Chaos, Solitons & Fractals, 2009

Let hðxÞ be a polynomial with real coefficients. We introduce hðxÞ-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these hðxÞ-Fibonacci polynomials. We also introduce hðxÞ-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q h ðxÞ that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers.

Generalized Fibonacci Polynomials and Fibonomial Coefficients

Annals of Combinatorics, 2014

The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s, t given by {0} = 0, {1} = 1, and {n} = s{n−1}+t{n−2} for n ≥ 2. The latter are defined by

Some Generalized Fibonacci Polynomials

2007

We introduce polynomial generalizations of the r-Fibonacci, r-Gibonacci, and r- Lucas sequences which arise in connection with two statistics defined, respectively, on linear, phased, and circular r-mino arrangements.

A new approach to generalized Fibonacci and Lucas numbers with binomial coefficients

Applied Mathematics and Computation, 2013

In this study, Fibonacci and Lucas numbers have been obtained by using generalized Fibonacci numbers. In addition, some new properties of generalized Fibonacci numbers with binomial coefficients have been investigated to write generalized Fibonacci sequences in a new direct way. Furthermore, it has been given a new formula for some Lucas numbers.

Generalized Fibonacci-Lucas Polynomials

International Journal of Advanced Mathematical Sciences, 2013

Various sequences of polynomials by the names of Fibonacci and Lucas polynomials occur in the literature over a century. The Fibonacci polynomials and Lucas polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, Generalized Fibonacci-Lucas Polynomials are introduced and defined by the recurrence relation

The generalized k-Fibonacci polynomials and generalized k-Lucas polynomials

Notes on Number Theory and Discrete Mathematics, 2021

In this paper, we define new families of Generalized Fibonacci polynomials and Generalized Lucas polynomials and develop some elegant properties of these families. We also find the relationships between the family of the generalized k-Fibonacci polynomials and the known generalized Fibonacci polynomials. Furthermore, we find new generalizations of these families and the polynomials in matrix representation. Then we establish Cassini’s Identities for the families and their polynomials. Finally, we suggest avenues for further research.

On the Properties of Generalized Fibonacci Like Polynomials

The Fibonacci polynomial has been generalized in many ways,some by preserving the initial conditions,and others by preserving the recurrence relation.In this article,we study new generalization fMng(x), with initial conditions M0(x) = 2 and M1(x) = m(x) + k(x), which is generated by the recurrence relation Mn+1(x) = k(x)Mn(x) + Mn􀀀1(x) for n  2, where k(x),m(x) are polynomials with real coefficients.We produce an extended Binet’s formula for fMng(x) and,thereby identities such as Simpson’s,Catalan’s,d’Ocagene’s,etc.using matrix algebra.Moreover, we present sum formulas concerning this new generalization.

Identities for the Generalized Fibonacci Polynomial

Integers, 2018

A second order polynomial sequence is of Fibonacci type (Lucas type) if its Binet formula is similar in structure to the Binet formula for the Fibonacci (Lucas) numbers. In this paper we generalize identities from Fibonacci numbers and Lucas numbers to Fibonacci type and Lucas type polynomials. A Fibonacci type polynomial is equivalent to a Lucas type polynomial if they both satisfy the same recurrence relations. Most of the identities provide relationships between two equivalent polynomials. In particular, each type of identities in this paper relate the following polynomial sequences: Fibonacci with Lucas, Pell with Pell-Lucas, Fermat with Fermat-Lucas, both types of Chebyshev polynomials, Jacobsthal with Jacobsthal-Lucas and both types of Morgan-Voyce.

Generalized Fibonacci-Lucas Sequence

Turkish Journal of Analysis and Number Theory, 2014

The Fibonacci sequence is a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula − − = + , 2 n ≥ with B 0 = 2b, B 1 = s, where b and s are integers. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet's formula and other simple methods.